Why do observers in special relativity see different measurements?

[b2386]

Hi guys,

I am in a class in intro physics and am having trouble fully understanding the theory of special relativity. A couple of "apparent" contradictions encountered in my textbook's explanations seem to be throwing me off. (These are based off the famous train example)
Observer 1: on track
Observer 2: on train

1) Observer 1 says that 2's clock goes slow, and observer 2 says that 1's clock goes slow.

2) Observer 1 says that 2's meter sticks are contracted along their direction of relative motion, and observer 2 says that 1's meter sticks are contracted.

These 2 observations seem to be contradictory. Can someone please explain to me why they are not?

 

[selfAdjoint]

They seem contradictory to you because you are (perhaps unconsciously) adopting a "perferrred frame" in which you can see both A and B "simultanoeously" and determine "what really happens". All of those scare quotes are there to gently hint that this is not the way to think about relativity. A is in her own world (called a rest frame) and B is in his. And YOU are in yours, no better than either of theirs, and subjected to the fact that you are NOT allowed to extend your local physics to them because they are in motion relative to you (unless you enter into the frame of one or the other in imagination - you cannnot really do both at once, and be physical), as well as of each other.

So the only way A can experience B's measurements or B experience A's, is by electromagnetic interactions - light in this case. And what they see, the only physics they can share in this simplified case, is that each of them sees the the other's measures transformed from her own. The relationship is symmetrical because relativity doesn't know or care which observer is "really" moving - the platform could just as well have been speeding away from th train which is standing still.

 

[JustinLevy]

Hmm... it sounds like you can probably do the math, but it still doesn't make sense as to why it is not a contradiction. Correct? In that case I may be able to help.

Let's start with these two (apparently contradictory) claims:
1) Observer 1 says that 2's clock goes slow
2) Observer 2 says that 1's clock goes slow


The problem here is that the two claims are NOT talking about the same events. If they were talking about the same pair of events and observer 1 says observer 2 measured less time between the events AND observer 2 says observer 1 measured less time between the events, then yes, this would be a REAL paradox. But as stated, this is not the case here.

Try this out, pick any two random events (coordinate points) according to observer 1. What is the time between the events according to observer 1? Transform to observer 2's frame. What is the time between events according to observer 2? Transfer back to observer 1's frame as well if you want to feel more confident about the math (you should indeed get back the original event coordinants). Hopefully this will demonstrate to you that if observer 1 and 2 are talking about the same events, they will agree which one measured less time. Stop and think for a moment about the math you just did and why this must always be true.

So the "trick" to these "train paradoxes" is that the observers aren't talking about the same events when they make such claims. In the case of the clocks, for example, observer 1 measures the time between two events that are BOTH at the position of observer 2 (the "moving" observer according to observer 1, and yes the "moving" observer will see less time between these events). To have observer 2 make the same claim, they need to use a DIFFERENT set of events (that now are both at the position of observer 1). If this change in events being discussed isn't made explicit, it can accidentally "appear" like a paradox (when of course there isn't one).

I hope that helped some. Good luck in your studies.